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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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Chapter 7 Electromagnetism

7.1Introduction

The electromagnetic force is central to nearly every physical process around us and is a major component of classical physics. In fact, the development of electromagnetic theory in the nineteenth century gave us much mathematical machinery that we now apply quite generally in other ﬁelds, including potential theory, vector calculus, and the ideas of divergence and curl.

It is therefore not surprising that this section deals with a large array of physical quantities and their relationships. As usual, SI units are assumed throughout. In the past electromagnetism has su ered from the use of a variety of systems of units, including the cgs system in both its electrostatic (esu) and electromagnetic (emu) forms. The fog has now all but cleared, but some specialised areas of research still cling to these historical measures. Readers are advised to consult the section on unit conversion if they come across such exotica in the literature.

Equations cast in the rationalised units of SI can be readily converted to the once common Gaussian (unrationalised) units by using the following symbol transformations:

Equation conversion: SI to Gaussian units

0 →1/(4π)	µ0 →4π/c2	B →B/c
χE →4πχE	χH →4πχH	H →cH /(4π)
A →A/c	M →cM	D →D/(4π)

The quantities ρ, J , E , φ, σ, P , r, and µr are all unchanged.

136	Electromagnetism

7.2 Static ﬁelds

Electrostatics

Electrostatic

E = − φ

electric ﬁeld

(7.1)

electrostatic

potential

φa − φb = a

= − b

φa

potential at a

Potential

E · dl

φb

potential at b

di erencea

(7.2)

line element

charge density

Poisson’s Equation

φ =

−

(7.3)

permittivity of

(free space)

free space

φ(r) =

(7.4)

Point charge at r

4π 0|r − r |

point charge

E (r) =

q(r − r )

(7.5)

4π 0|r − r |3

dτ

Field from a

E (r) =

ρ(r )(r − r )

dτ

(7.6)

dτ

volume element

charge distribution

position vector

(free space)

4π 0

−

of dτ

volume

aBetween points a and b along a path l.

Magnetostaticsa

Magnetic scalar

φm

magnetic scalar

B = −µ0 φm

(7.7)

potential

magnetic ﬂux

density

φm in terms of the

solid angle of a

φm =

IΩ

(7.8)

Ω

loop solid angle

generating current

4π

current

loop

line element in

Biot–Savart law (the

µ0I

−

r )

the direction of

ﬁeld from a line

B(r) =

(7.9)

the current

line

−

current)

position vector of

4π

current density

Ampere’s`

law

×B = µ0J

(7.10)

µ0

permeability of

(di erential form)

free space

Ampere’s` law (integral

B · dl = µ0Itot

(7.11)

tot

total current

form)

through loop

aIn free space.

7.2 Static ﬁelds	137

Capacitancea

Of sphere, radius a

C = 4π 0 ra

(7.12)

Of circular disk, radius a

C = 8 0 ra

(7.13)

Of two spheres, radius a, in

C = 8π 0 raln2

(7.14)

contact

Of circular solid cylinder,

C [8 + 4.1(l/a)

0.76

] 0 ra

(7.15)

radius a, length l

Of nearly spherical surface,

3.139

10−11

rS1/2

(7.16)

area S

Of cube, side a

7.283

10−11

(7.17)

Between concentric spheres,

C = 4π 0 rab(b

−

a)−1

(7.18)

radii a < b

Between coaxial cylinders,

C =

2π 0 r

per unit length

(7.19)

radii a < b

ln(b/a)

C =

π 0 r

per unit length

(7.20)

Between parallel cylinders,

arcosh(d/a)

separation 2d, radii a

π 0 r

(d a)

(7.21)

ln(2d/a)

Between parallel, coaxial

0 rπa2

circular disks, separation d,

+ 0 ra[ln(16πa/d) − 1]

(7.22)

radii a

aFor conductors, in an embedding medium of relative permittivity r.

Inductancea

Of N-turn solenoid

L = µ0N2A/l

(straight or toroidal),

(7.23)

length l, area A ( l2)

Of coaxial cylindrical

µ0

tubes, radii a, b (a < b)

L =

per unit length

(7.24)

2π

Of parallel wires, radii a,

µ0

per unit length, (2d a)

(7.25)

separation 2d

Of wire of radius a bent in

a loop of radius b a

L µ0b ln

− 2

(7.26)

aFor currents conﬁned to the surfaces of perfect conductors in free space.

138	Electromagnetism

Electric ﬁeldsa

E (r) =

(r < a)

Uniformly charged sphere,

radius a, charge q

4πq0

(7.27)

r (r

4π

Uniformly charged disk,

q 0

≥

radius a, charge q (on axis,

E (z) =

−

√

2π 0a2

|z|

z2 + a2

(7.28)

Line charge, charge density

E (r) =

(7.29)

λ per unit length

2π 0r2

Electric dipole, moment p

Er =

pcosθ

(7.30)

(spherical polar

2π 0r3

coordinates, θ angle

Eθ =

psinθ

(7.31)

between p and r)

4π 0r3

Charge sheet, surface

E =

(7.32)

density σ

2 0

aFor r = 1 in the surrounding medium.

−	θ	+
	p

Magnetic ﬁeldsa

Uniform inﬁnite solenoid,

µ0nI inside (axial)

current I, n turns per unit

B = "0

outside

(7.33)

length

Uniform cylinder of

µ0Ir/(2πa )

r < a

(7.34)

current I, radius a

B(r) = "µ0I/(2πr)

≥

Magnetic dipole, moment

Br = µ0

mcosθ

(7.35)

2πr3

m (θ angle between m and

µ0msinθ

Bθ =

(7.36)

4πr3

Circular current loop of N

µ0NI

B(z) =

(7.37)

turns, radius a, along axis, z

(a2 + z2)3/2

The axis, z, of a straight

µ0nI

solenoid, n turns per unit

Baxis =

(cosα1 − cosα2)

(7.38)

length, current I

aFor µr = 1 in the surrounding medium.

θ m

α2 α1 z

Image charges

Real charge, +q, at a distance:	image point	image charge

b from a conducting plane	−b	−q
b from a conducting sphere, radius a	a2/b	−qa/b
b from a plane dielectric boundary:
seen from free space	−b	−q( r − 1)/( r + 1)
seen from the dielectric	b	+2q/( r + 1)

7.3 Electromagnetic ﬁelds (general)	139

7.3 Electromagnetic ﬁelds (general)

Field relationships

Conservation of

∂ρ

current density

charge density

charge

· J = − ∂t

(7.39)

time

Magnetic vector

B = ×A

(7.40)

vector potential

potential

Electric ﬁeld from

∂A

−

electrical

potentials

E = −

∂t

(7.41)

potential

Coulomb gauge

· A = 0

(7.42)

condition

Lorenz gauge

1 ∂φ

speed of light

condition

· A + c2

∂t = 0

(7.43)

1 ∂2φ

dτ

Potential ﬁeld

c2 ∂t2

−

φ = 0

(7.44)

equationsa

1 ∂2A

A = µ0J

(7.45)

c2 ∂t2

−

Expression for φ

φ(r,t) =

ρ(r ,t− |r − r |/c) dτ

(7.46)

dτ

volume element

position vector of

in terms of ρa

4π 0

−

dτ

volume

Expression for A

A(r,t) = µ0

J (r

,t− |r − r |/c) dτ

(7.47)

µ0

permeability of

in terms of J a

4π

−

free space

volume

aAssumes the Lorenz gauge.

Lienard´–Wiechert potentialsa
			q	charge
Electrical potential of		q	r	vector from charge
Electrical potential of	φ =	q	(7.48)	to point of
	φ =		(7.48)	to point of
a moving point charge		4π 0(\|r\| − v · r/c)		observation
			v	particle velocity
Magnetic vector		µ0qv		q
potential of a moving	A =	µ0qv	(7.49)	v
				v
		4π(\|r\| − v		r
point charge		4π(\|r\| − v	· r/c)	r

aIn free space. The right-hand sides of these equations are evaluated at retarded times, i.e., at t = t− |r |/c, where r is the vector from the charge to the observation point at time t .

140 Electromagnetism

Maxwell’s equations

Di erential form:

Integral form:

E =

(7.50)

· ds =

ρ dτ

(7.51)

closed surface

volume

B = 0

(7.52)

B · ds = 0

(7.53)

closed surface

×E = −

∂B

loop E · dl = −

dΦ

(7.55)

(7.54)

∂t

∂E

dl = µ0I + µ0 0

∂E

ds (7.57)

×B = µ0J + µ0 0 ∂t

(7.56)

loop

surface

∂t

Equation (7.51) is “Gauss’s law”

surface element

Equation (7.55) is “Faraday’s law”

dτ

volume element

electric ﬁeld

line element

magnetic ﬂux density

linked magnetic ﬂux (=

ds)

current density

linked current (=

ds%)

charge density

time

Maxwell’s equations (using D and H )

Di erential form:

Integral form:

D = ρfree

(7.58)

D · ds =

ρfree dτ

(7.59)

closed surface volume

B = 0

(7.60)

B · ds = 0

(7.61)

closed surface

×E = −

∂B

loop E · dl = −

dΦ

(7.63)

(7.62)

∂t

∂D

(7.65)

×H = J free +

∂t

(7.64)

loop H · dl =

free surface

∂t ·

displacement ﬁeld

electric ﬁeld

ρfree

free charge density (in the sense of

surface element

ρ = ρinduced + ρfree)

dτ

volume element

magnetic ﬂux density

line element

magnetic ﬁeld strength

linked magnetic ﬂux (=

B · ds)

J free

free current density (in the sense of

free linked free current (=

ds)

%free

J = J induced + J free)

time

7.3 Electromagnetic ﬁelds (general)	141

Relativistic electrodynamics

electric ﬁeld

Lorentz

= E

(7.66)

magnetic ﬂux density

measured in frame moving

transformation of

= γ(E + v

(7.67)

at relative velocity v

electric and

= B

(7.68)

Lorentz factor

]−

1/2

magnetic ﬁelds

= γ(B

−

E /c2)

(7.69)

= [1 − (v/c)

parallel to v

perpendicular to v

Lorentz

ρ = γ(ρ

−

vJ /c2)

(7.70)

transformation of

= J

(7.71)

current density

current and charge

charge density

densities

J = γ(J

−

vρ)

(7.72)

Lorentz

φ = γ(φ− vA )

(7.73)

electric potential

transformation of

= A

(7.74)

magnetic vector potential

potential ﬁelds

= γ(A − vφ/c2)

(7.75)

J = (ρc,J )

(7.76)

A =

(7.77)

current density four-vector

Four-vector ﬁeldsa

potential four-vector

∂2

,− 2

(7.78)

D’Alembertian operator

∂t2

2A = µ0J

(7.79)

aOther sign conventions are common here. See page 65 for a general deﬁnition of four-vectors.

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